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Calculations of neutron diffraction patterns obtained with a nematic liquid crystal
J. Phys. Chem.Solids, 1973,Vol.34, pp. 735-747. PergamonPress. Printedin GreatBritain
CALCULATIONS OF NEUTRON DIFFRACTION PATTERNS OBTAINED WITH A NEMATIC LIQUID CRYSTAL R. PYNN
Institutt for Atomenergi, Kjeller, Norway (Received 3 July 1972)
Abstract- Recently, neutron diffraction data obtained with a fully deuterated sample of the nematic liquid crystal para-azoxyanisole (PAA) were reported. In the present paper a number of simple ideas which contribute to an understanding of these data are presented. It is shown that part of the structure of the observed diffraction patterns can be attributed to the wavelength dependence of a molecular form-factor. Further, it is found that diffuse scattering observed in P A A just below its melting point may be explained in terms of oscillations of molecules about their long axes. The temperature dependence of this diffuse scattering suggests that the melting of PAA may be driven by a soft torsional mode or modes. The explanation of the diffuse scattering from the solid and the similarities between the diffraction patterns of solid and nematic phases suggest that scattering from the nematic phase might be explicable in terms of hindered rotations of molecules about their long axes. A simple model based on this hypothesis and on the neglect of short-range orientational correlations of the molecular long axes is proposed. A comparison of results obtained from this model with the experimental data demonstrates that there is considerable short-range ordering of the long molecular axes in both the nematic and isotropic-liquid phases of PAA. 1. I N T R O D U C T I O N
Thermotropic liquid crystals are conveniently divided into three classes only one of which will be of interest in this paper. This class, known as nematic, is characterised by the molecular alignment mentioned above and by a complete lack of long-range positional ordering of molecules. Thus the simplest picture of a nematic which one may usefully keep in mind is that of a fluid of long rods which tend to be aligned in such a way that the fluid has overall uniaxial symmetry [2]. Various techniques may be employed to measure the degree of molecular alignment in a nematic[1,3], but, apart from this information, very little is known about the positional and orientational correlations of the molecules. X-ray measurements have been made [4, 5] but, in the majority of cases, the analysis of these data has contributed little to our detailed understanding of the liquidcrystalline state. The reason for this lack of progress is fairly clear: the liquid-crystalline
THE LIQUID-CRYSTALLINEstate of matter[l] is essentially an intermediate state whose individual physical properties are often shared with either the solid or isotropic-liquid states. Those liquid crystals which are composed of one type of molecule are said to be thermotropic and occur in a temperature region which is bounded below by the solid state and above by the liquid state. Thermotropic liquid crystals are composed of elongated, rectilinear molecules whose long axes tend to be spontaneously aligned. Although the alignment constitutes a long-range correlation of molecular orientations, a macroscopic specimen is generally composed of 'domains' whose directions of preferred orientation are unrelated. However, the application of an external perturbation, such as a magnetic field, is often sufficient to cause a mutual alignment of the 'domains' and to produce a macroscopic specimen with a single preferred direction. 735
736
R. P Y N N
state presents all the problems characteristic of a liquid plus the complication of anisotropy. Recently, measurements have been made at this laboratory of the neutron-diffraction properties of a fully deuterated sample of the nematic, para-azoxyanisole (PAA). These results[6] (Ref. [6] is hereafter referred to as I) are partially reproduced in Fig. 1. Parts B, C and D of this figure were obtained in the nematic phase at 119~ for three relative orientations of the neutron scattering vector Q and an applied magnetic field, H. Figures 1B and 1D were obtained with Q and H respectively perpendicular and parallel to one another while Fig. IC depicts data for I
I
I
I
f
H = 0. Since, for PAA, the direction of preferred molecular alignment (optic axis) is parallel to H, Figs. 1B and 1D contain information about (nuclear) density correlations perpendicular and parallel to the optic axis respectively. In Fig. 1E data for the isotropic liquid (at 129~ is shown while, in Fig. 1A, a spectrum obtained just below the melting point of P A A is displayed. The data of Fig. 1 are, with the exception of Fig. 1A, fairly similar to corresponding results obtained with X-rays. The explanation of Fig. 1 thus represents the challenge to which this paper responds and, as will become apparent, Fig. 1A is the key which will allow at least some progress to be made. 29 NEUTRON DIFFRACTION CROSS-SECTION FOR A MOLECULAR SYSTEM
i
I III
I
I
Ir
A :soft solid ll4~
-
} Nemafic 1190C E :lsotropic 1290C
:6# :.y---: , " :"":~ 0
.'.:%"
to m o~ 4m
I I
"":~':':"
A "t.:.. ..~176 -9 ~ .. ;. B
""
It is convenient to begin this discussion by writing down the diffraction cross-section for neutrons scattered by a target system composed of several nuclear species. If it is assumed that the scattering nuclei behave as Boltzmann particles, the cross-section may be written as a generalisation of Van Hove's expression [7]. Thus one may write: do"
"9
..:....
""~'.* ~ : ::. ...t. 9
dO = ~ [a~~176 +
~" ."-'.:.'....
mn
:.'~,:.'f-'.r "~""
""~"'."~'.:".~A:<"":
""
.'.,r :.~;" ~,
. . ,~L~9..-.:.,r ~.. 9
0
-
. ...':
...~: :.'."
0
I 20
X:~
E
I 30
Scattering
"'h~.~
I 40
angle,
9
50
60
deg
Fig. 1. Neutron diffraction patterns recorded with deuterated PAA. Part B - Q _L H: part D -- Q II H: parts C and E - - H = 0. Part A was obtained with solid P A A after the sample had been field-cooled in the Q • H configuration.
m Z
~mn(aine) ] (e-iO'r" eiO'*')r" (2.1)
In this equation ae~'h and a.ll'lC n' are, respectively, the bound coherent and incoherent scattering lengths of the mth nucleus and rm is the instantaneous position of this nucleus: ( . . . . . . . >r denotes a thermal average. In its present form equation (2.1) does not provide a particularly convenient description of the neutron scattering from a molecular system9 The reason for this is that positional correlations of nuclei which belong to the same molecule are not displayed explicitly in (2.1) which thus gives an atomic rather than a molecular description of the scattering. The remedy for this defect is clearly to replace the rm by combinations of coordinates which
C A L C U L A T I O N S OF N E U T R O N D I F F R A C T I O N PATTERNS
describe molecular and intramolecular positions. Such a transformation was first given by Zemach and Glauber[8] who made the assumption that all molecular and (normal) intramolecular coordinates are dynamically independent. Even under such restrictive conditions the expression which replaces (2. l) is inconveniently complicated, basically because the internal molecular coordinates are kinematically coupled to those coordinates which describe molecular orientation. In addition one may be fairly certain that the neglect of molecular position-orientation correlations is inadequate for liquid crystals. While this latter objection to the ZemachGlauber formalism is not difficult to circumvent, the only way in which one may overcome the complication of the kinematic coupling of internal and orientational degrees of freedom is by assuming molecules to be rigid9 One may advance several excuses, based on physical arguments, in support of the use of this assumption in the present application. However, the unpleasant truth of the matter is that any more realistic assumption quickly leads to a calculation which is intractable and, for this reason, the rigidmolecule approximation will be adopted here without further apology. Within the context of the rigid-molecule approximation one may write rm = Ri + D(a',fliTi) " Uj
rotated through the appropriate Euler angles. Substituting (2.2) in (2.1) gives be
do"
do-\
do-\
a--g -
(2.3a)
where d_~) = ~
~ [ a J . a j , " +Sjj,(ai~)z ]
s
i
COil
COD
.0'
9 ( e-iQ'D''u'eiQ'n''uj' )T
(2.3b)
and
do')a = ~ ~ t
(a
aj as' coh
ir
iQS,U~eiQD.....
eohx-
Jr'
9e -iQR' e i~
)r.
(2.3c)
In equations (2.3), do-/dgl)s and do-/dll)d represent, respectively, scattering from a single molecule and from distinct molecules and Di is an abbreviation for D(aifliyi). ZZ'
X ~
\ i
(2.2)
Z"Z m
Z u
where the nuclear index rn has been separated into an index i which refers to a molecule and an index j which labels a particular nucleus within the molecule. In equation (2.2) R~ is the position of a molecular centre-ofmass and (a~fliyt) are tile Euler angles (we use the convention given by Rose [9] which, for ease of reference is reproduced in Fig. 2) which carry the laboratory coordinate system into the principal-axis (body) coordinate system of the l~h molecule9 The matrix D of equation (2.2) causes the intramolecular coordinate uj (defined in the body system) to
737
III
=
~x"
\y~'"
X~
Fig. 2. Definition of the Euler angles used in this paper. The long molecular axis coincides with z" and the optic axis of the sample is in the z direction. 3. SCATTERING BY A NEMATIC: SOME SIMPLE IDEAS
In order to make further progress in a calculation of scattering by a nematic, a
738
R. P Y N N
specific form for the matrices Di is required. The simplest approximation one may make rests on the assumption that the alignment of molecules in a nematic is nearly perfect and that Di describes infinitesimal deviations from this aligned state. One may then define unit vectors which are parallel to the molecular axes and write D, in terms of the firstorder changes in these vectors. Such a theory is simple to work out and gives a cross-section which, not surprisingly, is very similar to that given by Lubensky [2] for light scattering. In spite of its appealing simplicity, however, the infinitesimal-fluctuation theory is of such limited applicability that it is, for practical purposes, useless for neutron work. The reason for this is easily appreciated if one examines typical values of the order parameter, S, of nematics. This quantity, defined by S = (P~(cos/3i)),
where P2 is a Legendre polynomial, lies typically between 0.3 and 0.7 for a nematic. Thus fluctuations of the long molecular axes through angles of 30 ~ and 40 ~ from the preferred direction are by no means unlikely. This fact makes nonsense of an infinitesimaldeviation theory in all cases except those for which Q is extremely small. In such an instance the scattered neutrons average molecular fluctuations over a large volume of the target system and 'see' only small effective fluctuations of orientation. This is, of course, the case for light scattering experiments. One may try to extend the infinitesimalfluctuation theory in several ways. As a first guess one might introduce something akin to spin-wave theory[10] and take approximate account of the finite angular deviation of a molecule from perfect alignment. However, such an approach is, at best, only a crude approximation which will break down for values of S which are not unusual for nematics. Clearly, such a theory is totally inadequate in the isotropic liquid phase.
Another approach to this problem is to use a theory which has been propounded in the context of N M R measurements with liquid crystals[11]. Here the idea is to define a local axis of alignment which is subject to infinitesimal fluctuations and to allow individual molecules to reorientate about the local axis. The disadvantage of this model, as it is presently applied, is that correlations between molecular coordinates are ignored except in so far as they contribute to coherent fluctuations of the local preferred axis. It is difficult to believe that such a theory would adequately describe neutron diffraction data but the author confesses to not having worked out the detailed consequences of such an approach. 4. SINGLE M O L E C U L E SCATTERING
Since the simple theories which have been discussed above are not applicable to neutron scattering by nematics one is obliged to seek more sophisticated alternatives. One part of the cross-section given by (2.3a) is immediately amenable to a more general treatment than has yet been considered. This part, the 'self' or single-molecule scattering, (d~/ dl~)s, is in fact the dominant contribution to scattering at large Q as may be seen by examination of equations (2.3b) and (2.3c). This fact led de Gennes [ 12] to suggest that scattering experiments ought to be performed at large Q in order to render analysis of results straightforward. Unfortunately the data of Figure 1 were not taken exclusively at large Q. Nevertheless, part of the scattering is due to single molecule effects and, for this reason an expression for this contribution will now be developed. The analysis which follows is essentially the same as that given by de Gennes [ 12] and similar to that given by Sears[13]. Readers familiar with either of these papers are invited to proceed immediately to equation (4.6) which gives a final result which differs only in minor ways from that given by de Gennes.
CALCULATIONS
OF
NEUTRON
The first step in the analysis is to apply the usual Rayleigh expansion to the exponential term of (2.3b). Thus one writes
e -i~
47r ~] (i)~]t(Qv~,) /=0
rrt
9 Y'{,,(OQ,chQ)Ytm(Oj~',4)~') (4.1)
DIFFRACTION
PATTERNS
739
system. Physically this means that if any molecule is turned about its long axis the resuiting molecular configuration is exactly as likely as the original one. With this assumption the 7~ average in (4.3) may be carried out to give
( e-iQ'vjj')a~,= 47r ~ (t)~,(Qvz,) Y~o(O.,O) /=0
where v!J3t = Di. (uj--uj,) and Ivjj,[ = vjj,.
• D10*(0/3i0) l?10(0Jj',0)"
(4.4)
In equation (4.1) 0Q and ~bo are the polar and One may now make use of the relations azimuthal angles of Q while 0jj, and ~bz are the corresponding quantities for vj ~" 4~- 1/2y, j , " Ytm(O,~b) D~to(~ = (21+ l) im(fl, Or) is a spherical harmonic and jr(x) a spherical Bessel function. The arguments of the spheriand cal harmonics in (4.1) are to be evaluated in the laboratory coordinate system. However, r,o(O,O) = {2l [ ~ 1+ l']l/2pl(cos 0) this coordinate system is related to the body system of the ith molecule by the Euler angles to obtain (ad3iy~) and one may therefore write (Ref. [9],
p. 59)
(e-iQ'*~')T= ~ (i)ljt(Qvjj,)Pt(cos OQ)(21-t- 1). /=0
Ylm( Ojj%ff)jj,) = Z Dram, I* (~ me
Ylm' (Oz' ,ffgjj,) (4.2)
where Elm is to be evaluated with 0z, and (b~j, defined in the body coordinate system and D mm' ~ is a rotation matrix. Equation (4.2) may be substituted in equation (4.1) and, in view of the uniaxial symmetry of the nematic, the resulting expression may be averaged with respect to ~. This gives
(e-i~
i = 4Ir ~, (t)~it(Qv~j,)Y~o(Oo,~bo) /=0
• Y~ D~*m,(Ofl(yOf/tm,(Ojj,,q~jj,). (4.3) /?l p
In order to simplify this expression still further one may make an assumption which will be shown later in this paper to be reasonable and which is additionally supported by N M R measurements[14]. This assumption is that the variable y~ is not correlated with any other dynamical coordinate of the nematic
X/Sl(cos O~j,)(Pt(cos flO)r-
(4.5)
Equation (4.5) may be further simplified by noting that a nematic is symmetric about a mirror plane perpendicular to the optic axis. This fact restricts the sum over l in (4.5) to even integers and gives, finally
acohacoh ~, (-- 1)(/2t(Qvj~,) JJ t
/=0
P2t(COS0o) (41+ 1) • (P21(cos ri0)r+ (ai~c)zSjj,1.
0z,) (4.6)
As noted above, this expression differs only slightly from that given by de Gennes[12]: the (-- 1)~ factor in (4.6) replaces de Gennes' (-- 1)2~ factor and, in addition, (4.6) is simpler than de Gennes' result because the average over y~ has already been performed in a physically plausible manner. Our excuse for
740
R. P Y N N
repeating the derivation of (4.6) is that it makes Section 8 of this paper less cumbersome. 5. M E A N - F I E L D
THEORY
The evaluation of equation (4.6) is straightforward once the quantity (P21(COS/3i))T has been determined and thus, in this section, a simple method of obtaining the average of P2t(cos/3i) will be discussed. Before beginning this discussion it is perhaps well to recollect that/34 is the angle between the long axis of the z~h molecule and the optic axis of the nematic sample (cf. Fig. 2). In this context the 'long axis' of a molecule is that principal axis about which the molecule has least moment of inertia. Thus one sees that, for a perfectly aligned nematic, all P2z(cos/3i) are unity since /3, = 0: in this case the sum on l of equation (4.6) may be performed exactly to give a result involving (cylindrical) Bessel functions. On the other hand, for an isotropic liquid (P2t (cosBi)) is zero for l # 0. In the nematic phase one expects to obtain something between these two extremes. In order to find (P2t(cos/3i) )r an expression for the statistical distribution of the /34 is required. Let us suppose that n(/3)sin/3d/3 is the probability of finding a molecule inclined to the optic axis at an angle between /3 and /3 + dB. Then, clearly
between the centres of mass of molecules 1 and 2. The mean potential energy, E, of molecule 2 (say) may now be found by averaging the interaction energy given above with respect to the polar coordinates /31 and al of molecule 1. This procedure is consistent with ignoring the short-range interaction between molecules 1 and 2 and gives kBT oc --
f da~ 0
sin/31 d/31 Y~ Y2~m(/31Otl) 0
m
Y2m(132az) n(/31) = --A S
P2(cos/32) (5.2)
where A is a (density dependent) constant and S is the order parameter defined earlier. From equation (5.2) one may immediately write n(fl) = exp [.4 S P2(cos/3)]/ 0
exp [,4 S Pz(cos/3)] sin/3d/3.
(5.3)
In order to determine the constant A of equation (5.3) a self consistency requirement may be introduced: if P2(cos/3) is averaged over the distribution given by (5.3) the result should be equal to S. After some algebra this is found to imply that
x/-a
(P2z(cos flO)r = f n(fl)P21(cos/3) sin/3d/3 0
(5.1)
where a = 3 A S/2
if n(/3) is suitably normalised. To find n(/3) one may resort to the mean-field approximation of Maier and Saupe[15]. In this approximation it is assumed that the long-range interaction potential of a pair of molecules is of dispersive character and may be written as A' 47r
R6 5 ~
Y*(/3,a,)Y2m(/32az)
where A' is a constant and R is the distance
andD(a) is Dawson's integral defined by D(ot) = e -~2 f eX2dx. 0
Since the solutions of equation (5.4) may be readily tabulated the problem of determining n(/3) has been solved. However, for the evaluation of equation (4.6) it is convenient to expand n(/3) in the form
CALCULATIONS
OF NEUTRON
DIFFRACTION
n(/3) = • ckPk(cos/3).
60m
k
In this case one finds from the orthogonality relations for the Legendre polynomials that the first few useful ce are given by Co= 1 : c2 = 5S :
C4= ~9(,0, - - 14 T) 3
6. EVALUATION OF SINGLE MOLECULE SCATTERING
In terms of the coefficients ck introduced at the end of the last section, equation (4.6) may be rewritten as = N s
jy
X
(--
ac~h cola /=0
l)~21(Qvjj,)
I I
~
I
\
~
74l
1
I
I
I
I
40-E
-
7,
zo-
13
--
-~
o
~' c o
60 --
~
-
~
x~
40--
\
m 03
l
t
.E
I k,5,_-O. 8
I " QIIH
/~\
\\,.
20--
\
~~/~,111
X\ \
k
~,
0
do-
PATTERNS
I
I
I
I
I
I
Lo
20
30
40
50
60
scottering
ongle,
70
deg
Fig. 3. Coherent 'single molecule' scattering computed for deuterated P A A from equation (4.6).
P21(COS 0o) c2lP2t(COS 0jj,)
+ (ain~c)zaj~,].
(4.6a)
This equation has been evaluated for deuterated PAA on the assumption that the molecular conformation is identical to that found in the solid phase of normal, hydrated PAA [ 16]. No attempt has been made to ensure the convergence of (4.6a): rather the sum on l has been truncated after the l = 2 term. As an excuse for this procedure one may note that the mean-field distribution n(/3) given in Section 5 is certainly incorrect because it entirely ignores short-range correlations of molecular orientations. Thus there is nothing to be gained by faithfully reproducing this distribution in the evaluation of (4.6a). Results obtained for the coherent part of the cross-section given by (4.6a) are displayed in Fig. 3. It is immediately clear from this figure that the broad peak at a scattering angle, 20, of approximately 55 ~ in Fig. 1C and 1D has been obtained in the calculation. In addition, the absence of the 55 ~ peak in Fig. 1B is also explained. Thus this peak corresponds to intramolecular correlations and would occur in Fig. IC and 1D even in the absence of correlations between different molecules.
In contrast, the peak at 20 - 30 ~ in Fig. 1B, C and E is completely absent from the curves of Fig. 3 and thus cannot be explained on the basis of the simple ideas presented so far. The results shown in Fig. 3 indicate that a measurement of the intensity of the 20 - 55~ peak as a function of temperature would provide direct information about the temperaturedependence of the nematic order parameter, S. 7. SCATTERING IN THE SOFT-SOLID REGION
The failure of the simple ideas so far discussed to explain the data of Fig, 1 leaves one in somewhat of a quandry. Clearly the explanation must lie in the 'distinct' term of equation (2.3). However, this term is, in general, not straightforwardly calculated and it is therefore useful to reexamine the data of Fig. 1 to see if any simple procedure is suggested by these data. One fact which is clear from Fig. 1 is the apparently continuous manner in which the nematic diffraction pattern of Fig. I B develops from the pattern displayed in Fig. 1A. This latter figure displays data obtained with a field-cooled solid at a temperature slightly below the melting point: in I this state of the
742
R, P Y N N
sample was designated the soft-solid region. While, in view of the first-order nature of the melting transition, the assertion of a continuous development of Fig. 1B from Fig. 1A cannot be completely correct, the data imply that a partial explanation of Fig. 1B may result from a consideration of Fig. 1A. In fact, Fig. IA suggests that the soft-solid region should be relatively easily understood. The presence of sharp Bragg peaks in this figure confirms that the sample is indeed solid in the sense that long-range correlations between molecular centres-of-mass are maintained. Further, experience suggests that the broad peak at 20 - 30 ~ cannot be explained simply in terms of phonons in the lattice. Thus a little thought shows that the only remaining plausible degree of freedom is that which represents reorientations of molecules, or subgroups thereof, about the long molecular axes. In order to make a more quantitative analysis of the diffraction from the soft solid one may rewrite the coherent part of equation (2.3a) in the form do" d---~ : ~
e-iQl~ eiOR" l<~
+ N <}a(Q){2>r
(7.1)
where
tional motions of the coordinate/3: in this case the symbol ( . . . . . >r may be replaced by an average over the third Euler angle, 3'. Let us now assume that some subgroups, comprising nuclei 1,2 . . . . . . m, are fixed while other subgroups, comprising nuclei ( m + 1 ) . . . n , are free to reorientate about the long molecular axis. Since we deal only with diffraction data (i.e. data which are not energy analysed) the model for this reorientation is of no importance. The only quantity which is needed is the probability distribution of the variable 3': for simplicity this is assumed to be independent of the value of 3". The final manipulation of (7.1) which is required before the evaluation is performed is to separate the scattering into Bragg scattering plus a remainder which we shall call the diffuse scattering. Thus one writes do-
m
dfl
e--iQ'R'eiQ'~'I(~(Q)>T{~ nil i anti i'
+ N [(la(Q) 12>~-{ (a(Q))rl z] (7.2) where the first term is the Bragg scattering and the second is the additional, diffuse scattering which arises because nuclei ( m + l) . . . . n are free to reorientate about the long molecular axis. Evaluation of (7.2) is relatively straightforward and one finds eventually that
a(Q) = ~ a J e -iQ'5,'u, coh
j=l
In writing equation (7.1) it has been assumed that the coordinates specifying molecular orientation are independent of those specifying molecular position and that orientation coordinates of different molecules are uncorrelated. Thus the temperature averages in (7.1) average the molecular form-factor or(Q) over possible orientations of each molecule. For simplicity, it is assumed in (7.1) that the molecular centres-of-mass are located on a rigid lattice and thus no temperature averaging over ~ is required. To further simplify (7.1) let us ignore libra-
(I a(Q)[~>~- {(~(Q)>~{ ~ =
aja j,j'=m
r' cob cola COS (Q3U35"~3 X [ J 0 ( O j j')
- Jo ( Q • p~)J o( Q • pJ') ] .
(7.3)
In this equation Qz and Q~_ are components of Q parallel and perpendicular to the long molecular axis, Uzjj' and p~' are corresponding components of (us-us,) and pJ is the perpendicular component of u: the functions J0 are zero-order Bessel functions. Although equation (7.3) has been deduced on the basis of some extremely drastic as-
C A L C U L A T I O N S OF N E U T R O N D I F F R A C T I O N PATTERNS
sumptions it may nevertheless contain an explanation of the data displayed in Fig. 1A. That this is so may be seen from Fig. 4 in which are plotted results obtained from equation (7.3) when either the whole molecule or just the CD3 groups are allowed to reorientate. The diffuse intensity obtained for molecular reorientation displays a considerable resemblance to that actually measured, a fact which indicates that the soft-solid region may be a sort of one-dimensional plastic crystal. One 50
t
[
w
I
/
\
/
Soft
\
/
o
[
\
/
40 m
I
\
solid
PAA
_
\ \
0
50
\
--
\ \ \
4-
PAA 20
molecule
--
k
0
N
743
but which retains the mathematical form of the plastic-crystal model. The simplest model which satisfies the criteria discussed above is based on the assumption that each molecule is an Einstein oscillator with the Euler angle 3' as the harmonically varying coordinate. Thus one visualises each molecule as executing torsional oscillations about the long molecular axis. In this situation the average over T of equation (7.2) involves the (Gaussian) distribution function characteristic of a harmonic oscillator. In addition, however, a further average over the most probable values of,/is required* since the soft-solid sample used in I was polycrystalline in the plane perpendicular to the optic axis. After some algebra one finds that these considerations lead to a diffuse scattering (the scattering which occurs in addition to the Bragg peaks) given by
--
CH3_O_ ~ _ N = N _ ~ _ O _ C H3 "-...... diffuse
jj, oo
o
I0
20
Scoffering
50
angle,
I
I
I
40
50
60
1 / dye_~,~jo(Q• 70
deg
Fig. 4. Coherent, diffuse scattering from the soft-solid region of deuterated PAA computed from equation (7.3): ...... molecular reorientations: reorientations of CD3 groups only.
notices also from Fig. 4 that rotation of only CDa groups is insufficient to explain the observed scattering from the soft solid. The model described above has one unfortunate defect: it does not permit a natural interpretation of the temperature dependence of the soft-solid scattering described in I. It was found in I that, in the 5~ region in which PAA is a soft-solid, the intensity of the broad peak at 2 0 - 30 ~ increases monotonically with temperature. To explain this variation within the framework of a plastic-crystallike model would be difficult if not impossible. Thus one is obliged to seek a model which involves temperature as a natural parameter
--2Ojp~,Xcos (Tjj,+2o-y)]l/2)]. (7.4) In this equation TJJ, is the angle between OJ and Or' while o- is the angular width of the molecular oscillations, that is
l r' kB
where is Boltzmann's constant, ~oE the Einstein frequency of the molecular oscillations and 1 is the smallest molecular moment of inertia. Equation (7.4) has two pleasant features: it vanishes for large ~oe and gives the plastic*For convenience, the most probable values of y are assumed to be the same for each molecule in a given crystallite. In particular, equation (7.1) cannot be used unless this assumption is made.
744
R. PYNN
crystal result in the limit of small tot. In spite of its appearance, equation (7.4) is easy to evaluate numerically by a Hermite integration method. The result of such an evaluation for several values of tot is displayed in Fig. 5 which shows that the essential features of the temperature variation of the soft-solid scattering may be obtained in terms of an Einstein frequency which becomes softer with increasing temperature. 50
t
I
J
I
I
I
~ue=0 . 8 2 x IOI3~ec-I E
40--
.r
L~ 3 0 - -
ZO--
~o -I03
o
to
20 3o 4o 5o Scattering angle, deg
60
TO
Fig. 5. Coherent, diffuse scattering from the soft-solid region of deuterated PAA computed from equation (7.4) for various values of the Einstein frequency at a temperature of 300 K.
While it is possible to couple the molecular oscillations described above by using a random-phase approximation, there is little to be gained by such an approach because the intermolecular potential is not known. Nevertheless, the foregoing calculations provide a plausible explanation of the qualitative properties of the soft solid. In particular, it seems likely that the 20 - 30~ diffuse peak in Fig. 1A arises because molecules oscillate through considerable angles about their long axes. Further, the calculation indicates that melting to the nematic phase may be driven by a soft-mode behaviour and that, in the nematic phase, hindered rotations of molecules about their long axes are to be expected.
We shall return to a discussion of the melting of liquid crystals in a forthcoming experimental paper [17]. 8. 'MOLECULARLY INCOHERENT' SCATTERING
The relative success of the preceding calculation in explaining the diffraction pattern of the soft solid and the similarities between this pattern and that of the nematic (Figs. 1A and 1B) suggests that one should attempt to describe soft-solid and nematic phases in a similar manner. Two essential approximations seem to be involved in the calculation of scattering from the soft solid: the centre of mass and reorientational motions of molecules are assumed to be uncoupled and the molecules are assumed to remain in a state of perfect alignment. Clearly the latter postulate cannot be correct in the nematic, a fact which suggests that one should investigate the consequences of the former assumption. A simple way in which this may be implemented is by assuming that the (classical) molecular pairdistribution function may be written as g(l~--Ri,) n(fl~) n(flv) where g(Ri--Ri,) is an unspecified function of the relative positions of molecules i and i' and n(fl) was given in Section 5. Thus, one assumes that if an arbitrarily chosen molecule has position and orientation coordinates (R~aJ3iy~) at a given point in time then the probability of finding another molecule at (Rvo~vflvTv) at the same instant is just g(Ri-- Re) n(fli) n(flv) sin/34 sin fly. In the limit of large ( R i - Rv) this pair distribution function (pdf) is correct and the function g(Ri--Rv) may be assumed to account for the short-range positional correlations of molecules. However the suggested pdf ignores entirely the short-range orientational correlation of molecules. The pdf suggested above may be used to work out the 'distinct molecule' part of the scattering cross-section given by equation (2.3c). The analysis is very similar to that given in Section 4, the major difference being the number of symbols which have to be carried at each stage. From equation (2.3c) one
CALCULATIONS
OF NEUTRON
has, in terms of the a(Q) defined in equation (7.1):
= ~, (e-iQa'eiQ'a")r I(a(Q)>..~l = (8.1a) i#i t
= I S ( Q ) - 1] I (o,(Q)>~l 2
(8. lb)
DIFFRACTION
PATTERNS
745
to understand why the S(Q) term of (8.1b) may not contribute greatly to the structure of Fig. 1B. The product of S(Q) and the molecular form factor (plotted in Fig. 6) will, because of the forms of these functions, tend to be a small weakly oscillating contribution to the scattering. One may see this even more
where
80 6r
S(Q) = 1 + p f daR [g(R) -- 1] e -iQ.lt (8.1 c)
I
~\
I
r.
I
l
I
50
60
and P is the molecular number d e n s i t y . Analysis similar to that of Section 4 yields ~ 4c ve3 v
(ct(Q))~ = ~ a eoh J J
( - 1)Zc2d2z(Quj)
~o
/=0
• P2t(cos 0Q)P2t(cos 0j)
\
(8.2)
where 0j is the polar angle of uj in the body coordinate system. In view of the form assumed for the p d f the similarity of (8. lb) and the 'distinct' contribution to the scattering from the soft solid (equation (7.2)) is not entirely surprising. The differences between the two expressions lie in the changed definition of the molecular form factor (a(Q)) and in the replacement of the soft-solid Bragg scattering by the molecular structure factor S(Q) in the nematic. One may safely assume that S(Q) has a form typical for the liquid state: it will be small at low Q, increase to a peak value of a little more than unity and finally converge to unity at large Q. One may use the above observations, in conjunction with the patterns displayed in Figs. 1A and 1B, to argue that the S(Q) term in (8.1b) is not of major importance in an explanation of the first broad peak in Fig. lB. This peak occurs, in the soft solid, in addition to Bragg scattering and, in view of the experimental fact that Fig. 1B appears to grow more or less continuously from Fig. 1A, one might expect a similar explanation of this peak to pertain in both the soft-solid and nematic phases. It is, in addition, fairly easy
0
I0
20 30 40 Scatl-ering angle, deg
70
Fig. 6. Squared molecular form-factor I (a(Q)) I= for the nematic phase of deuteratcd PAA. The scale on the ordinate is the same as for Figs. 3, 4, 5, and 7.
clearly in the case of a hypothetical fluid of freely rotating dumbell molecules: in this case the first node of the molecular form factor corresponds exactly to the position of the first peak in S(Q) and the rapid variation of both of these functions at low Q is cancelled when their product is formed. The above arguments suggest that the part of the diffraction crosssection which is responsible for the observed structure at 2 0 - 30 ~ in Fig. 1B may be written as do
d---~ ~ (I~(Q) I2>~*-I<~(Q))"~I=
(8.3)
The first term in this equation is identical with (do-/df~)~ and is thus given by equation (4.6) while the second term may be found by squaring equation (8.2). In view of its similarity to the usual expression for incoherent scattering [7], one might term equation (8.3) the molecularly-incoherent approximation. Physically this approximation means that neutrons 'see' correlations between positions of nuclei be-
746
R. P Y N N
longing to a particular molecule but that no correlations of coordinates of different molecules are included. Thus (8.3) involves the 'single molecule' scattering plus that part of the 'distinct molecule' scattering which is independent of S(Q). The advantage of equation (8.3) is that it essentially apes many of the mathematical features of the diffuse cross-section obtained for the soft solid and might therefore be expected to reproduce the structure of Fig. lB. Furthermore, the derivation of equation (8.3) implies that it can only be seriously deficient if there are strong short-range correlations between values of the Euler angle/3 for different molecules. Thus one may regard the comparison of equation (8.3) with experiment as a test of the hypothesis that short-range orientational correlations are weak. Results obtained from equation (8.3) for deuterated PAA are shown in Fig. 7. From this figure it is apparent that the discrepancy 60 /
I
I
I
I
I
V
,.,,
= o
HJ.O
/ 0
/
L...r ~
I
I
I
\.
I
I
60
~-
~QIH ----QIIH
4o
.s
I
o
I
I
I
A
2
I
I
-
_
I
s:o.o
_
20
--
I 0
I0
20
30
Scott"ering angle,
40
50
60
70
deg
Fig. 7. Molecularly-incoherent scattering for the nematic phase of deuterated P A A computed from equation (8.3).
between experiment and calculation is most marked for small values of the order parameter S. Thus, only when long-range orientational order is large enough to account for part of the short-range ordering do results obtained from equation (8.3) approximate the observed diffraction pattern. Equivalently one may say that the neutron diffraction data demonstrate clearly the existence of shortrange orientational order in both nematic and isotropic-liquid phases. Thus the route to a more sophisticated explanation of neutron diffraction by nematics now seems clear. One may be reasonably cavalier in treating short-range positional correlations of the molecules but must find an adequate treatment of the short-range orientational correlations. If this can be done, a moment's thought about the results of the soft-solid calculations indicates that one should be able to explain both the nematic diffraction patterns and the temperature dependence of these patterns presented in I. It is hoped that this will form the subject of a future paper. 9. CONCLUSIONS The purpose of this paper has been to investigate the information which is contained in neutron diffraction patterns obtained with nematic samples. Since no theory has been found which gives a complete description of these patterns the paper is mainly a collection of attempts to discover, by trial and error, the important contributions to the diffraction patterns. In this context it has been shown that part of the structure of the diffraction patterns obtained with nematic PAA can be explained solely on the basis of 'single molecule' scattering. Thus, the position of the 20 - 55 ~diffuse peak in Figs. 1C, D and E reflects the intramolecular structure of PAA while the intensity of this peak is determined by the degree of nematic ordering and the relative orientation of the neutron scattering vector and the optic axis of the sample.
C A L C U L A T I O N S OF N E U T R O N D I F F R A C T I O N P A T T E R N S
Perhaps the major success of this paper is the explanation of the diffuse scattering which arises in the soft-solid region and which is displayed in Fig. 1A. This diffuse scattering appears to arise because molecules perform large torsional oscillations about their long axes. Although oscillations of neighbouring molecules are certainly coupled the model used here ignores this coupling. This leads to an expression for the diffuse scattering which has been termed the molecularly-incoherent approximation and which may be applied with minor modifications to the nematic phase. The failure of this model to describe scattering from the fluid phases of P A A clearly demonstrates the large degree of short-range molecular alignment which exists in both nematic and isotropic-liquid phases and which is ignored in the molecularly-incoherent approximation. Thus one concludes that the diffraction patterns obtained with deuterated PAA certainly contain information about orientational correlations of molecules and that the patterns cannot be explained unless these correlations are taken into account. A cknowledgements- I would like to thank the Norwegian Council for Scientific and Industrial Research for financial support and lnstitun for Atomenergi for their hospitality during 1971-72. My special thanks are due to
747
Dr. T. Riste for introducing me to the problems discussed in this paper and for many stimulating discussions concerning the contents of the paper. REFERENCES 1. A recent review, particularly useful for its reference list, has been written by: BROWN G. H., D O A N E J. W. and N E F F V. D., A Review of the Structure and Physical Properties of Liquid Crystals', Butterworths, London ( 1971). 2. The possibility of biaxiality has been most recently discussed by L U B E N S K Y T. C., Phys. Rev. A2, 2497 (1970). 3. R O W E L L J. C., P H I L L I P S W. D., MELBY L. R. and P A N A R M., J. Chem. Phys. 43, 3422 (1965). 4. F A L G U E I R E T T E S J., Bull. Soc. Franc. Miner. Crist. 82, 171 (1959). 5. C H I S T Y A K O V I. G, and C H A I K O V S K I ! V. M., Soviet Phys. Crystallogr 12, 770 (1968). 6. P Y N N R., O T N E S K. and RISTE T., Solid State Commun. 11, 1365-1367 (1972). 7. V A N H O V E L., Phys. Rev. 95, 249 (1954). 8. Z E M A C H A. C. and G L A U B E R R. J., Phys. Rev. 101, 129 (1956). 9. ROSE M, E., Elementary Theory of Angular Momentum, Wiley, New York (1957). 10. K E F F E R F. and L O U D O N R., J. appl. Phys. 32, Suppl., 2S (1961). 11. D O A N E J. W. and J O H N S O N D. L., Chem. Phys. Lett. 6, 291 (1970). 12. DE G E N N E S P. G., Comp. Rend. 274B, 142 (1972). 13. SEARS V. F., Can. J. Phys. 44, 1299 (1966). 14. L I P P M A N N H., Ann. Phys. Leipzig 20, 265 (1957). 15. M A I E R W. and S A U P E A., Z. Naturf 14A, 882 (1959); Z. Naturf. 15A, 287 (1960). 16. K R I G B A U M W. R., C H A T A N I Y. and BARBER P. G., Acta crystallogr. B26, 97 (1970). 17. RISTE T. and P Y N N R., Solid State Commun. (in press).
CALCULATIONS OF NEUTRON DIFFRACTION PATTERNS OBTAINED WITH A NEMATIC LIQUID CRYSTAL R. PYNN
Institutt for Atomenergi, Kjeller, Norway (Received 3 July 1972)
Abstract- Recently, neutron diffraction data obtained with a fully deuterated sample of the nematic liquid crystal para-azoxyanisole (PAA) were reported. In the present paper a number of simple ideas which contribute to an understanding of these data are presented. It is shown that part of the structure of the observed diffraction patterns can be attributed to the wavelength dependence of a molecular form-factor. Further, it is found that diffuse scattering observed in P A A just below its melting point may be explained in terms of oscillations of molecules about their long axes. The temperature dependence of this diffuse scattering suggests that the melting of PAA may be driven by a soft torsional mode or modes. The explanation of the diffuse scattering from the solid and the similarities between the diffraction patterns of solid and nematic phases suggest that scattering from the nematic phase might be explicable in terms of hindered rotations of molecules about their long axes. A simple model based on this hypothesis and on the neglect of short-range orientational correlations of the molecular long axes is proposed. A comparison of results obtained from this model with the experimental data demonstrates that there is considerable short-range ordering of the long molecular axes in both the nematic and isotropic-liquid phases of PAA. 1. I N T R O D U C T I O N
Thermotropic liquid crystals are conveniently divided into three classes only one of which will be of interest in this paper. This class, known as nematic, is characterised by the molecular alignment mentioned above and by a complete lack of long-range positional ordering of molecules. Thus the simplest picture of a nematic which one may usefully keep in mind is that of a fluid of long rods which tend to be aligned in such a way that the fluid has overall uniaxial symmetry [2]. Various techniques may be employed to measure the degree of molecular alignment in a nematic[1,3], but, apart from this information, very little is known about the positional and orientational correlations of the molecules. X-ray measurements have been made [4, 5] but, in the majority of cases, the analysis of these data has contributed little to our detailed understanding of the liquidcrystalline state. The reason for this lack of progress is fairly clear: the liquid-crystalline
THE LIQUID-CRYSTALLINEstate of matter[l] is essentially an intermediate state whose individual physical properties are often shared with either the solid or isotropic-liquid states. Those liquid crystals which are composed of one type of molecule are said to be thermotropic and occur in a temperature region which is bounded below by the solid state and above by the liquid state. Thermotropic liquid crystals are composed of elongated, rectilinear molecules whose long axes tend to be spontaneously aligned. Although the alignment constitutes a long-range correlation of molecular orientations, a macroscopic specimen is generally composed of 'domains' whose directions of preferred orientation are unrelated. However, the application of an external perturbation, such as a magnetic field, is often sufficient to cause a mutual alignment of the 'domains' and to produce a macroscopic specimen with a single preferred direction. 735
736
R. P Y N N
state presents all the problems characteristic of a liquid plus the complication of anisotropy. Recently, measurements have been made at this laboratory of the neutron-diffraction properties of a fully deuterated sample of the nematic, para-azoxyanisole (PAA). These results[6] (Ref. [6] is hereafter referred to as I) are partially reproduced in Fig. 1. Parts B, C and D of this figure were obtained in the nematic phase at 119~ for three relative orientations of the neutron scattering vector Q and an applied magnetic field, H. Figures 1B and 1D were obtained with Q and H respectively perpendicular and parallel to one another while Fig. IC depicts data for I
I
I
I
f
H = 0. Since, for PAA, the direction of preferred molecular alignment (optic axis) is parallel to H, Figs. 1B and 1D contain information about (nuclear) density correlations perpendicular and parallel to the optic axis respectively. In Fig. 1E data for the isotropic liquid (at 129~ is shown while, in Fig. 1A, a spectrum obtained just below the melting point of P A A is displayed. The data of Fig. 1 are, with the exception of Fig. 1A, fairly similar to corresponding results obtained with X-rays. The explanation of Fig. 1 thus represents the challenge to which this paper responds and, as will become apparent, Fig. 1A is the key which will allow at least some progress to be made. 29 NEUTRON DIFFRACTION CROSS-SECTION FOR A MOLECULAR SYSTEM
i
I III
I
I
Ir
A :soft solid ll4~
-
} Nemafic 1190C E :lsotropic 1290C
:6# :.y---: , " :"":~ 0
.'.:%"
to m o~ 4m
I I
"":~':':"
A "t.:.. ..~176 -9 ~ .. ;. B
""
It is convenient to begin this discussion by writing down the diffraction cross-section for neutrons scattered by a target system composed of several nuclear species. If it is assumed that the scattering nuclei behave as Boltzmann particles, the cross-section may be written as a generalisation of Van Hove's expression [7]. Thus one may write: do"
"9
..:....
""~'.* ~ : ::. ...t. 9
dO = ~ [a~~176 +
~" ."-'.:.'....
mn
:.'~,:.'f-'.r "~""
""~"'."~'.:".~A:<"":
""
.'.,r :.~;" ~,
. . ,~L~9..-.:.,r ~.. 9
0
-
. ...':
...~: :.'."
0
I 20
X:~
E
I 30
Scattering
"'h~.~
I 40
angle,
9
50
60
deg
Fig. 1. Neutron diffraction patterns recorded with deuterated PAA. Part B - Q _L H: part D -- Q II H: parts C and E - - H = 0. Part A was obtained with solid P A A after the sample had been field-cooled in the Q • H configuration.
m Z
~mn(aine) ] (e-iO'r" eiO'*')r" (2.1)
In this equation ae~'h and a.ll'lC n' are, respectively, the bound coherent and incoherent scattering lengths of the mth nucleus and rm is the instantaneous position of this nucleus: ( . . . . . . . >r denotes a thermal average. In its present form equation (2.1) does not provide a particularly convenient description of the neutron scattering from a molecular system9 The reason for this is that positional correlations of nuclei which belong to the same molecule are not displayed explicitly in (2.1) which thus gives an atomic rather than a molecular description of the scattering. The remedy for this defect is clearly to replace the rm by combinations of coordinates which
C A L C U L A T I O N S OF N E U T R O N D I F F R A C T I O N PATTERNS
describe molecular and intramolecular positions. Such a transformation was first given by Zemach and Glauber[8] who made the assumption that all molecular and (normal) intramolecular coordinates are dynamically independent. Even under such restrictive conditions the expression which replaces (2. l) is inconveniently complicated, basically because the internal molecular coordinates are kinematically coupled to those coordinates which describe molecular orientation. In addition one may be fairly certain that the neglect of molecular position-orientation correlations is inadequate for liquid crystals. While this latter objection to the ZemachGlauber formalism is not difficult to circumvent, the only way in which one may overcome the complication of the kinematic coupling of internal and orientational degrees of freedom is by assuming molecules to be rigid9 One may advance several excuses, based on physical arguments, in support of the use of this assumption in the present application. However, the unpleasant truth of the matter is that any more realistic assumption quickly leads to a calculation which is intractable and, for this reason, the rigidmolecule approximation will be adopted here without further apology. Within the context of the rigid-molecule approximation one may write rm = Ri + D(a',fliTi) " Uj
rotated through the appropriate Euler angles. Substituting (2.2) in (2.1) gives be
do"
do-\
do-\
a--g -
(2.3a)
where d_~) = ~
~ [ a J . a j , " +Sjj,(ai~)z ]
s
i
COil
COD
.0'
9 ( e-iQ'D''u'eiQ'n''uj' )T
(2.3b)
and
do')a = ~ ~ t
(a
aj as' coh
ir
iQS,U~eiQD.....
eohx-
Jr'
9e -iQR' e i~
)r.
(2.3c)
In equations (2.3), do-/dgl)s and do-/dll)d represent, respectively, scattering from a single molecule and from distinct molecules and Di is an abbreviation for D(aifliyi). ZZ'
X ~
\ i
(2.2)
Z"Z m
Z u
where the nuclear index rn has been separated into an index i which refers to a molecule and an index j which labels a particular nucleus within the molecule. In equation (2.2) R~ is the position of a molecular centre-ofmass and (a~fliyt) are tile Euler angles (we use the convention given by Rose [9] which, for ease of reference is reproduced in Fig. 2) which carry the laboratory coordinate system into the principal-axis (body) coordinate system of the l~h molecule9 The matrix D of equation (2.2) causes the intramolecular coordinate uj (defined in the body system) to
737
III
=
~x"
\y~'"
X~
Fig. 2. Definition of the Euler angles used in this paper. The long molecular axis coincides with z" and the optic axis of the sample is in the z direction. 3. SCATTERING BY A NEMATIC: SOME SIMPLE IDEAS
In order to make further progress in a calculation of scattering by a nematic, a
738
R. P Y N N
specific form for the matrices Di is required. The simplest approximation one may make rests on the assumption that the alignment of molecules in a nematic is nearly perfect and that Di describes infinitesimal deviations from this aligned state. One may then define unit vectors which are parallel to the molecular axes and write D, in terms of the firstorder changes in these vectors. Such a theory is simple to work out and gives a cross-section which, not surprisingly, is very similar to that given by Lubensky [2] for light scattering. In spite of its appealing simplicity, however, the infinitesimal-fluctuation theory is of such limited applicability that it is, for practical purposes, useless for neutron work. The reason for this is easily appreciated if one examines typical values of the order parameter, S, of nematics. This quantity, defined by S = (P~(cos/3i)),
where P2 is a Legendre polynomial, lies typically between 0.3 and 0.7 for a nematic. Thus fluctuations of the long molecular axes through angles of 30 ~ and 40 ~ from the preferred direction are by no means unlikely. This fact makes nonsense of an infinitesimaldeviation theory in all cases except those for which Q is extremely small. In such an instance the scattered neutrons average molecular fluctuations over a large volume of the target system and 'see' only small effective fluctuations of orientation. This is, of course, the case for light scattering experiments. One may try to extend the infinitesimalfluctuation theory in several ways. As a first guess one might introduce something akin to spin-wave theory[10] and take approximate account of the finite angular deviation of a molecule from perfect alignment. However, such an approach is, at best, only a crude approximation which will break down for values of S which are not unusual for nematics. Clearly, such a theory is totally inadequate in the isotropic liquid phase.
Another approach to this problem is to use a theory which has been propounded in the context of N M R measurements with liquid crystals[11]. Here the idea is to define a local axis of alignment which is subject to infinitesimal fluctuations and to allow individual molecules to reorientate about the local axis. The disadvantage of this model, as it is presently applied, is that correlations between molecular coordinates are ignored except in so far as they contribute to coherent fluctuations of the local preferred axis. It is difficult to believe that such a theory would adequately describe neutron diffraction data but the author confesses to not having worked out the detailed consequences of such an approach. 4. SINGLE M O L E C U L E SCATTERING
Since the simple theories which have been discussed above are not applicable to neutron scattering by nematics one is obliged to seek more sophisticated alternatives. One part of the cross-section given by (2.3a) is immediately amenable to a more general treatment than has yet been considered. This part, the 'self' or single-molecule scattering, (d~/ dl~)s, is in fact the dominant contribution to scattering at large Q as may be seen by examination of equations (2.3b) and (2.3c). This fact led de Gennes [ 12] to suggest that scattering experiments ought to be performed at large Q in order to render analysis of results straightforward. Unfortunately the data of Figure 1 were not taken exclusively at large Q. Nevertheless, part of the scattering is due to single molecule effects and, for this reason an expression for this contribution will now be developed. The analysis which follows is essentially the same as that given by de Gennes [ 12] and similar to that given by Sears[13]. Readers familiar with either of these papers are invited to proceed immediately to equation (4.6) which gives a final result which differs only in minor ways from that given by de Gennes.
CALCULATIONS
OF
NEUTRON
The first step in the analysis is to apply the usual Rayleigh expansion to the exponential term of (2.3b). Thus one writes
e -i~
47r ~] (i)~]t(Qv~,) /=0
rrt
9 Y'{,,(OQ,chQ)Ytm(Oj~',4)~') (4.1)
DIFFRACTION
PATTERNS
739
system. Physically this means that if any molecule is turned about its long axis the resuiting molecular configuration is exactly as likely as the original one. With this assumption the 7~ average in (4.3) may be carried out to give
( e-iQ'vjj')a~,= 47r ~ (t)~,(Qvz,) Y~o(O.,O) /=0
where v!J3t = Di. (uj--uj,) and Ivjj,[ = vjj,.
• D10*(0/3i0) l?10(0Jj',0)"
(4.4)
In equation (4.1) 0Q and ~bo are the polar and One may now make use of the relations azimuthal angles of Q while 0jj, and ~bz are the corresponding quantities for vj ~" 4~- 1/2y, j , " Ytm(O,~b) D~to(~ = (21+ l) im(fl, Or) is a spherical harmonic and jr(x) a spherical Bessel function. The arguments of the spheriand cal harmonics in (4.1) are to be evaluated in the laboratory coordinate system. However, r,o(O,O) = {2l [ ~ 1+ l']l/2pl(cos 0) this coordinate system is related to the body system of the ith molecule by the Euler angles to obtain (ad3iy~) and one may therefore write (Ref. [9],
p. 59)
(e-iQ'*~')T= ~ (i)ljt(Qvjj,)Pt(cos OQ)(21-t- 1). /=0
Ylm( Ojj%ff)jj,) = Z Dram, I* (~ me
Ylm' (Oz' ,ffgjj,) (4.2)
where Elm is to be evaluated with 0z, and (b~j, defined in the body coordinate system and D mm' ~ is a rotation matrix. Equation (4.2) may be substituted in equation (4.1) and, in view of the uniaxial symmetry of the nematic, the resulting expression may be averaged with respect to ~. This gives
(e-i~
i = 4Ir ~, (t)~it(Qv~j,)Y~o(Oo,~bo) /=0
• Y~ D~*m,(Ofl(yOf/tm,(Ojj,,q~jj,). (4.3) /?l p
In order to simplify this expression still further one may make an assumption which will be shown later in this paper to be reasonable and which is additionally supported by N M R measurements[14]. This assumption is that the variable y~ is not correlated with any other dynamical coordinate of the nematic
X/Sl(cos O~j,)(Pt(cos flO)r-
(4.5)
Equation (4.5) may be further simplified by noting that a nematic is symmetric about a mirror plane perpendicular to the optic axis. This fact restricts the sum over l in (4.5) to even integers and gives, finally
acohacoh ~, (-- 1)(/2t(Qvj~,) JJ t
/=0
P2t(COS0o) (41+ 1) • (P21(cos ri0)r+ (ai~c)zSjj,1.
0z,) (4.6)
As noted above, this expression differs only slightly from that given by de Gennes[12]: the (-- 1)~ factor in (4.6) replaces de Gennes' (-- 1)2~ factor and, in addition, (4.6) is simpler than de Gennes' result because the average over y~ has already been performed in a physically plausible manner. Our excuse for
740
R. P Y N N
repeating the derivation of (4.6) is that it makes Section 8 of this paper less cumbersome. 5. M E A N - F I E L D
THEORY
The evaluation of equation (4.6) is straightforward once the quantity (P21(COS/3i))T has been determined and thus, in this section, a simple method of obtaining the average of P2t(cos/3i) will be discussed. Before beginning this discussion it is perhaps well to recollect that/34 is the angle between the long axis of the z~h molecule and the optic axis of the nematic sample (cf. Fig. 2). In this context the 'long axis' of a molecule is that principal axis about which the molecule has least moment of inertia. Thus one sees that, for a perfectly aligned nematic, all P2z(cos/3i) are unity since /3, = 0: in this case the sum on l of equation (4.6) may be performed exactly to give a result involving (cylindrical) Bessel functions. On the other hand, for an isotropic liquid (P2t (cosBi)) is zero for l # 0. In the nematic phase one expects to obtain something between these two extremes. In order to find (P2t(cos/3i) )r an expression for the statistical distribution of the /34 is required. Let us suppose that n(/3)sin/3d/3 is the probability of finding a molecule inclined to the optic axis at an angle between /3 and /3 + dB. Then, clearly
between the centres of mass of molecules 1 and 2. The mean potential energy, E, of molecule 2 (say) may now be found by averaging the interaction energy given above with respect to the polar coordinates /31 and al of molecule 1. This procedure is consistent with ignoring the short-range interaction between molecules 1 and 2 and gives kBT oc --
f da~ 0
sin/31 d/31 Y~ Y2~m(/31Otl) 0
m
Y2m(132az) n(/31) = --A S
P2(cos/32) (5.2)
where A is a (density dependent) constant and S is the order parameter defined earlier. From equation (5.2) one may immediately write n(fl) = exp [.4 S P2(cos/3)]/ 0
exp [,4 S Pz(cos/3)] sin/3d/3.
(5.3)
In order to determine the constant A of equation (5.3) a self consistency requirement may be introduced: if P2(cos/3) is averaged over the distribution given by (5.3) the result should be equal to S. After some algebra this is found to imply that
x/-a
(P2z(cos flO)r = f n(fl)P21(cos/3) sin/3d/3 0
(5.1)
where a = 3 A S/2
if n(/3) is suitably normalised. To find n(/3) one may resort to the mean-field approximation of Maier and Saupe[15]. In this approximation it is assumed that the long-range interaction potential of a pair of molecules is of dispersive character and may be written as A' 47r
R6 5 ~
Y*(/3,a,)Y2m(/32az)
where A' is a constant and R is the distance
andD(a) is Dawson's integral defined by D(ot) = e -~2 f eX2dx. 0
Since the solutions of equation (5.4) may be readily tabulated the problem of determining n(/3) has been solved. However, for the evaluation of equation (4.6) it is convenient to expand n(/3) in the form
CALCULATIONS
OF NEUTRON
DIFFRACTION
n(/3) = • ckPk(cos/3).
60m
k
In this case one finds from the orthogonality relations for the Legendre polynomials that the first few useful ce are given by Co= 1 : c2 = 5S :
C4= ~9(,0, - - 14 T) 3
6. EVALUATION OF SINGLE MOLECULE SCATTERING
In terms of the coefficients ck introduced at the end of the last section, equation (4.6) may be rewritten as = N s
jy
X
(--
ac~h cola /=0
l)~21(Qvjj,)
I I
~
I
\
~
74l
1
I
I
I
I
40-E
-
7,
zo-
13
--
-~
o
~' c o
60 --
~
-
~
x~
40--
\
m 03
l
t
.E
I k,5,_-O. 8
I " QIIH
/~\
\\,.
20--
\
~~/~,111
X\ \
k
~,
0
do-
PATTERNS
I
I
I
I
I
I
Lo
20
30
40
50
60
scottering
ongle,
70
deg
Fig. 3. Coherent 'single molecule' scattering computed for deuterated P A A from equation (4.6).
P21(COS 0o) c2lP2t(COS 0jj,)
+ (ain~c)zaj~,].
(4.6a)
This equation has been evaluated for deuterated PAA on the assumption that the molecular conformation is identical to that found in the solid phase of normal, hydrated PAA [ 16]. No attempt has been made to ensure the convergence of (4.6a): rather the sum on l has been truncated after the l = 2 term. As an excuse for this procedure one may note that the mean-field distribution n(/3) given in Section 5 is certainly incorrect because it entirely ignores short-range correlations of molecular orientations. Thus there is nothing to be gained by faithfully reproducing this distribution in the evaluation of (4.6a). Results obtained for the coherent part of the cross-section given by (4.6a) are displayed in Fig. 3. It is immediately clear from this figure that the broad peak at a scattering angle, 20, of approximately 55 ~ in Fig. 1C and 1D has been obtained in the calculation. In addition, the absence of the 55 ~ peak in Fig. 1B is also explained. Thus this peak corresponds to intramolecular correlations and would occur in Fig. IC and 1D even in the absence of correlations between different molecules.
In contrast, the peak at 20 - 30 ~ in Fig. 1B, C and E is completely absent from the curves of Fig. 3 and thus cannot be explained on the basis of the simple ideas presented so far. The results shown in Fig. 3 indicate that a measurement of the intensity of the 20 - 55~ peak as a function of temperature would provide direct information about the temperaturedependence of the nematic order parameter, S. 7. SCATTERING IN THE SOFT-SOLID REGION
The failure of the simple ideas so far discussed to explain the data of Fig, 1 leaves one in somewhat of a quandry. Clearly the explanation must lie in the 'distinct' term of equation (2.3). However, this term is, in general, not straightforwardly calculated and it is therefore useful to reexamine the data of Fig. 1 to see if any simple procedure is suggested by these data. One fact which is clear from Fig. 1 is the apparently continuous manner in which the nematic diffraction pattern of Fig. I B develops from the pattern displayed in Fig. 1A. This latter figure displays data obtained with a field-cooled solid at a temperature slightly below the melting point: in I this state of the
742
R, P Y N N
sample was designated the soft-solid region. While, in view of the first-order nature of the melting transition, the assertion of a continuous development of Fig. 1B from Fig. 1A cannot be completely correct, the data imply that a partial explanation of Fig. 1B may result from a consideration of Fig. 1A. In fact, Fig. IA suggests that the soft-solid region should be relatively easily understood. The presence of sharp Bragg peaks in this figure confirms that the sample is indeed solid in the sense that long-range correlations between molecular centres-of-mass are maintained. Further, experience suggests that the broad peak at 20 - 30 ~ cannot be explained simply in terms of phonons in the lattice. Thus a little thought shows that the only remaining plausible degree of freedom is that which represents reorientations of molecules, or subgroups thereof, about the long molecular axes. In order to make a more quantitative analysis of the diffraction from the soft solid one may rewrite the coherent part of equation (2.3a) in the form do" d---~ : ~
e-iQl~ eiOR" l<~
+ N <}a(Q){2>r
(7.1)
where
tional motions of the coordinate/3: in this case the symbol ( . . . . . >r may be replaced by an average over the third Euler angle, 3'. Let us now assume that some subgroups, comprising nuclei 1,2 . . . . . . m, are fixed while other subgroups, comprising nuclei ( m + 1 ) . . . n , are free to reorientate about the long molecular axis. Since we deal only with diffraction data (i.e. data which are not energy analysed) the model for this reorientation is of no importance. The only quantity which is needed is the probability distribution of the variable 3': for simplicity this is assumed to be independent of the value of 3". The final manipulation of (7.1) which is required before the evaluation is performed is to separate the scattering into Bragg scattering plus a remainder which we shall call the diffuse scattering. Thus one writes do-
m
dfl
e--iQ'R'eiQ'~'I(~(Q)>T{~ nil i anti i'
+ N [(la(Q) 12>~-{ (a(Q))rl z] (7.2) where the first term is the Bragg scattering and the second is the additional, diffuse scattering which arises because nuclei ( m + l) . . . . n are free to reorientate about the long molecular axis. Evaluation of (7.2) is relatively straightforward and one finds eventually that
a(Q) = ~ a J e -iQ'5,'u, coh
j=l
In writing equation (7.1) it has been assumed that the coordinates specifying molecular orientation are independent of those specifying molecular position and that orientation coordinates of different molecules are uncorrelated. Thus the temperature averages in (7.1) average the molecular form-factor or(Q) over possible orientations of each molecule. For simplicity, it is assumed in (7.1) that the molecular centres-of-mass are located on a rigid lattice and thus no temperature averaging over ~ is required. To further simplify (7.1) let us ignore libra-
(I a(Q)[~>~- {(~(Q)>~{ ~ =
aja j,j'=m
r' cob cola COS (Q3U35"~3 X [ J 0 ( O j j')
- Jo ( Q • p~)J o( Q • pJ') ] .
(7.3)
In this equation Qz and Q~_ are components of Q parallel and perpendicular to the long molecular axis, Uzjj' and p~' are corresponding components of (us-us,) and pJ is the perpendicular component of u: the functions J0 are zero-order Bessel functions. Although equation (7.3) has been deduced on the basis of some extremely drastic as-
C A L C U L A T I O N S OF N E U T R O N D I F F R A C T I O N PATTERNS
sumptions it may nevertheless contain an explanation of the data displayed in Fig. 1A. That this is so may be seen from Fig. 4 in which are plotted results obtained from equation (7.3) when either the whole molecule or just the CD3 groups are allowed to reorientate. The diffuse intensity obtained for molecular reorientation displays a considerable resemblance to that actually measured, a fact which indicates that the soft-solid region may be a sort of one-dimensional plastic crystal. One 50
t
[
w
I
/
\
/
Soft
\
/
o
[
\
/
40 m
I
\
solid
PAA
_
\ \
0
50
\
--
\ \ \
4-
PAA 20
molecule
--
k
0
N
743
but which retains the mathematical form of the plastic-crystal model. The simplest model which satisfies the criteria discussed above is based on the assumption that each molecule is an Einstein oscillator with the Euler angle 3' as the harmonically varying coordinate. Thus one visualises each molecule as executing torsional oscillations about the long molecular axis. In this situation the average over T of equation (7.2) involves the (Gaussian) distribution function characteristic of a harmonic oscillator. In addition, however, a further average over the most probable values of,/is required* since the soft-solid sample used in I was polycrystalline in the plane perpendicular to the optic axis. After some algebra one finds that these considerations lead to a diffuse scattering (the scattering which occurs in addition to the Bragg peaks) given by
--
CH3_O_ ~ _ N = N _ ~ _ O _ C H3 "-...... diffuse
jj, oo
o
I0
20
Scoffering
50
angle,
I
I
I
40
50
60
1 / dye_~,~jo(Q• 70
deg
Fig. 4. Coherent, diffuse scattering from the soft-solid region of deuterated PAA computed from equation (7.3): ...... molecular reorientations: reorientations of CD3 groups only.
notices also from Fig. 4 that rotation of only CDa groups is insufficient to explain the observed scattering from the soft solid. The model described above has one unfortunate defect: it does not permit a natural interpretation of the temperature dependence of the soft-solid scattering described in I. It was found in I that, in the 5~ region in which PAA is a soft-solid, the intensity of the broad peak at 2 0 - 30 ~ increases monotonically with temperature. To explain this variation within the framework of a plastic-crystallike model would be difficult if not impossible. Thus one is obliged to seek a model which involves temperature as a natural parameter
--2Ojp~,Xcos (Tjj,+2o-y)]l/2)]. (7.4) In this equation TJJ, is the angle between OJ and Or' while o- is the angular width of the molecular oscillations, that is
l r' kB
where is Boltzmann's constant, ~oE the Einstein frequency of the molecular oscillations and 1 is the smallest molecular moment of inertia. Equation (7.4) has two pleasant features: it vanishes for large ~oe and gives the plastic*For convenience, the most probable values of y are assumed to be the same for each molecule in a given crystallite. In particular, equation (7.1) cannot be used unless this assumption is made.
744
R. PYNN
crystal result in the limit of small tot. In spite of its appearance, equation (7.4) is easy to evaluate numerically by a Hermite integration method. The result of such an evaluation for several values of tot is displayed in Fig. 5 which shows that the essential features of the temperature variation of the soft-solid scattering may be obtained in terms of an Einstein frequency which becomes softer with increasing temperature. 50
t
I
J
I
I
I
~ue=0 . 8 2 x IOI3~ec-I E
40--
.r
L~ 3 0 - -
ZO--
~o -I03
o
to
20 3o 4o 5o Scattering angle, deg
60
TO
Fig. 5. Coherent, diffuse scattering from the soft-solid region of deuterated PAA computed from equation (7.4) for various values of the Einstein frequency at a temperature of 300 K.
While it is possible to couple the molecular oscillations described above by using a random-phase approximation, there is little to be gained by such an approach because the intermolecular potential is not known. Nevertheless, the foregoing calculations provide a plausible explanation of the qualitative properties of the soft solid. In particular, it seems likely that the 20 - 30~ diffuse peak in Fig. 1A arises because molecules oscillate through considerable angles about their long axes. Further, the calculation indicates that melting to the nematic phase may be driven by a soft-mode behaviour and that, in the nematic phase, hindered rotations of molecules about their long axes are to be expected.
We shall return to a discussion of the melting of liquid crystals in a forthcoming experimental paper [17]. 8. 'MOLECULARLY INCOHERENT' SCATTERING
The relative success of the preceding calculation in explaining the diffraction pattern of the soft solid and the similarities between this pattern and that of the nematic (Figs. 1A and 1B) suggests that one should attempt to describe soft-solid and nematic phases in a similar manner. Two essential approximations seem to be involved in the calculation of scattering from the soft solid: the centre of mass and reorientational motions of molecules are assumed to be uncoupled and the molecules are assumed to remain in a state of perfect alignment. Clearly the latter postulate cannot be correct in the nematic, a fact which suggests that one should investigate the consequences of the former assumption. A simple way in which this may be implemented is by assuming that the (classical) molecular pairdistribution function may be written as g(l~--Ri,) n(fl~) n(flv) where g(Ri--Ri,) is an unspecified function of the relative positions of molecules i and i' and n(fl) was given in Section 5. Thus, one assumes that if an arbitrarily chosen molecule has position and orientation coordinates (R~aJ3iy~) at a given point in time then the probability of finding another molecule at (Rvo~vflvTv) at the same instant is just g(Ri-- Re) n(fli) n(flv) sin/34 sin fly. In the limit of large ( R i - Rv) this pair distribution function (pdf) is correct and the function g(Ri--Rv) may be assumed to account for the short-range positional correlations of molecules. However the suggested pdf ignores entirely the short-range orientational correlation of molecules. The pdf suggested above may be used to work out the 'distinct molecule' part of the scattering cross-section given by equation (2.3c). The analysis is very similar to that given in Section 4, the major difference being the number of symbols which have to be carried at each stage. From equation (2.3c) one
CALCULATIONS
OF NEUTRON
has, in terms of the a(Q) defined in equation (7.1):
= ~, (e-iQa'eiQ'a")r I(a(Q)>..~l = (8.1a) i#i t
= I S ( Q ) - 1] I (o,(Q)>~l 2
(8. lb)
DIFFRACTION
PATTERNS
745
to understand why the S(Q) term of (8.1b) may not contribute greatly to the structure of Fig. 1B. The product of S(Q) and the molecular form factor (plotted in Fig. 6) will, because of the forms of these functions, tend to be a small weakly oscillating contribution to the scattering. One may see this even more
where
80 6r
S(Q) = 1 + p f daR [g(R) -- 1] e -iQ.lt (8.1 c)
I
~\
I
r.
I
l
I
50
60
and P is the molecular number d e n s i t y . Analysis similar to that of Section 4 yields ~ 4c ve3 v
(ct(Q))~ = ~ a eoh J J
( - 1)Zc2d2z(Quj)
~o
/=0
• P2t(cos 0Q)P2t(cos 0j)
\
(8.2)
where 0j is the polar angle of uj in the body coordinate system. In view of the form assumed for the p d f the similarity of (8. lb) and the 'distinct' contribution to the scattering from the soft solid (equation (7.2)) is not entirely surprising. The differences between the two expressions lie in the changed definition of the molecular form factor (a(Q)) and in the replacement of the soft-solid Bragg scattering by the molecular structure factor S(Q) in the nematic. One may safely assume that S(Q) has a form typical for the liquid state: it will be small at low Q, increase to a peak value of a little more than unity and finally converge to unity at large Q. One may use the above observations, in conjunction with the patterns displayed in Figs. 1A and 1B, to argue that the S(Q) term in (8.1b) is not of major importance in an explanation of the first broad peak in Fig. lB. This peak occurs, in the soft solid, in addition to Bragg scattering and, in view of the experimental fact that Fig. 1B appears to grow more or less continuously from Fig. 1A, one might expect a similar explanation of this peak to pertain in both the soft-solid and nematic phases. It is, in addition, fairly easy
0
I0
20 30 40 Scatl-ering angle, deg
70
Fig. 6. Squared molecular form-factor I (a(Q)) I= for the nematic phase of deuteratcd PAA. The scale on the ordinate is the same as for Figs. 3, 4, 5, and 7.
clearly in the case of a hypothetical fluid of freely rotating dumbell molecules: in this case the first node of the molecular form factor corresponds exactly to the position of the first peak in S(Q) and the rapid variation of both of these functions at low Q is cancelled when their product is formed. The above arguments suggest that the part of the diffraction crosssection which is responsible for the observed structure at 2 0 - 30 ~ in Fig. 1B may be written as do
d---~ ~ (I~(Q) I2>~*-I<~(Q))"~I=
(8.3)
The first term in this equation is identical with (do-/df~)~ and is thus given by equation (4.6) while the second term may be found by squaring equation (8.2). In view of its similarity to the usual expression for incoherent scattering [7], one might term equation (8.3) the molecularly-incoherent approximation. Physically this approximation means that neutrons 'see' correlations between positions of nuclei be-
746
R. P Y N N
longing to a particular molecule but that no correlations of coordinates of different molecules are included. Thus (8.3) involves the 'single molecule' scattering plus that part of the 'distinct molecule' scattering which is independent of S(Q). The advantage of equation (8.3) is that it essentially apes many of the mathematical features of the diffuse cross-section obtained for the soft solid and might therefore be expected to reproduce the structure of Fig. lB. Furthermore, the derivation of equation (8.3) implies that it can only be seriously deficient if there are strong short-range correlations between values of the Euler angle/3 for different molecules. Thus one may regard the comparison of equation (8.3) with experiment as a test of the hypothesis that short-range orientational correlations are weak. Results obtained from equation (8.3) for deuterated PAA are shown in Fig. 7. From this figure it is apparent that the discrepancy 60 /
I
I
I
I
I
V
,.,,
= o
HJ.O
/ 0
/
L...r ~
I
I
I
\.
I
I
60
~-
~QIH ----QIIH
4o
.s
I
o
I
I
I
A
2
I
I
-
_
I
s:o.o
_
20
--
I 0
I0
20
30
Scott"ering angle,
40
50
60
70
deg
Fig. 7. Molecularly-incoherent scattering for the nematic phase of deuterated P A A computed from equation (8.3).
between experiment and calculation is most marked for small values of the order parameter S. Thus, only when long-range orientational order is large enough to account for part of the short-range ordering do results obtained from equation (8.3) approximate the observed diffraction pattern. Equivalently one may say that the neutron diffraction data demonstrate clearly the existence of shortrange orientational order in both nematic and isotropic-liquid phases. Thus the route to a more sophisticated explanation of neutron diffraction by nematics now seems clear. One may be reasonably cavalier in treating short-range positional correlations of the molecules but must find an adequate treatment of the short-range orientational correlations. If this can be done, a moment's thought about the results of the soft-solid calculations indicates that one should be able to explain both the nematic diffraction patterns and the temperature dependence of these patterns presented in I. It is hoped that this will form the subject of a future paper. 9. CONCLUSIONS The purpose of this paper has been to investigate the information which is contained in neutron diffraction patterns obtained with nematic samples. Since no theory has been found which gives a complete description of these patterns the paper is mainly a collection of attempts to discover, by trial and error, the important contributions to the diffraction patterns. In this context it has been shown that part of the structure of the diffraction patterns obtained with nematic PAA can be explained solely on the basis of 'single molecule' scattering. Thus, the position of the 20 - 55 ~diffuse peak in Figs. 1C, D and E reflects the intramolecular structure of PAA while the intensity of this peak is determined by the degree of nematic ordering and the relative orientation of the neutron scattering vector and the optic axis of the sample.
C A L C U L A T I O N S OF N E U T R O N D I F F R A C T I O N P A T T E R N S
Perhaps the major success of this paper is the explanation of the diffuse scattering which arises in the soft-solid region and which is displayed in Fig. 1A. This diffuse scattering appears to arise because molecules perform large torsional oscillations about their long axes. Although oscillations of neighbouring molecules are certainly coupled the model used here ignores this coupling. This leads to an expression for the diffuse scattering which has been termed the molecularly-incoherent approximation and which may be applied with minor modifications to the nematic phase. The failure of this model to describe scattering from the fluid phases of P A A clearly demonstrates the large degree of short-range molecular alignment which exists in both nematic and isotropic-liquid phases and which is ignored in the molecularly-incoherent approximation. Thus one concludes that the diffraction patterns obtained with deuterated PAA certainly contain information about orientational correlations of molecules and that the patterns cannot be explained unless these correlations are taken into account. A cknowledgements- I would like to thank the Norwegian Council for Scientific and Industrial Research for financial support and lnstitun for Atomenergi for their hospitality during 1971-72. My special thanks are due to
747
Dr. T. Riste for introducing me to the problems discussed in this paper and for many stimulating discussions concerning the contents of the paper. REFERENCES 1. A recent review, particularly useful for its reference list, has been written by: BROWN G. H., D O A N E J. W. and N E F F V. D., A Review of the Structure and Physical Properties of Liquid Crystals', Butterworths, London ( 1971). 2. The possibility of biaxiality has been most recently discussed by L U B E N S K Y T. C., Phys. Rev. A2, 2497 (1970). 3. R O W E L L J. C., P H I L L I P S W. D., MELBY L. R. and P A N A R M., J. Chem. Phys. 43, 3422 (1965). 4. F A L G U E I R E T T E S J., Bull. Soc. Franc. Miner. Crist. 82, 171 (1959). 5. C H I S T Y A K O V I. G, and C H A I K O V S K I ! V. M., Soviet Phys. Crystallogr 12, 770 (1968). 6. P Y N N R., O T N E S K. and RISTE T., Solid State Commun. 11, 1365-1367 (1972). 7. V A N H O V E L., Phys. Rev. 95, 249 (1954). 8. Z E M A C H A. C. and G L A U B E R R. J., Phys. Rev. 101, 129 (1956). 9. ROSE M, E., Elementary Theory of Angular Momentum, Wiley, New York (1957). 10. K E F F E R F. and L O U D O N R., J. appl. Phys. 32, Suppl., 2S (1961). 11. D O A N E J. W. and J O H N S O N D. L., Chem. Phys. Lett. 6, 291 (1970). 12. DE G E N N E S P. G., Comp. Rend. 274B, 142 (1972). 13. SEARS V. F., Can. J. Phys. 44, 1299 (1966). 14. L I P P M A N N H., Ann. Phys. Leipzig 20, 265 (1957). 15. M A I E R W. and S A U P E A., Z. Naturf 14A, 882 (1959); Z. Naturf. 15A, 287 (1960). 16. K R I G B A U M W. R., C H A T A N I Y. and BARBER P. G., Acta crystallogr. B26, 97 (1970). 17. RISTE T. and P Y N N R., Solid State Commun. (in press).